Exact error linearization is a well-known full-order observer design method which yields linear time-invariant, stable error dynamics in normal form coordinates. The difficult procedure associated with exact error linearization has led to the simpler extended Luenberger observer design. However, extended Luenberger observers can only guarantee a first-order approximation to the exact error linearization design. This paper provides an explicit formula involving only derivative (Lie derivative and bracket) and matrix inversion operations for computing an Nth-order approximation of the exact error linearization observer. Hence, the proposed method is a computational simple tool for designing observers for systems admitting exact error linearization observers. Two examples demonstrate the application of the method.